Problem 116 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The sum of -18 and \(-6,\) divided by the product of 2 and -4

Step-by-Step Solution

Answer

3

Step 1: Understand the Problem

The task is to write a numerical expression for the given phrase and then simplify it. The phrase is: 'The sum of -18 and -6, divided by the product of 2 and -4'.

Step 2: Write the Expression for the Sum

First, find the sum of -18 and -6. This can be written as \[-18 + (-6) = -18 - 6\].

Step 3: Simplify the Sum

Now simplify the expression for the sum: \[-18 - 6 = -24\].

Step 4: Write the Expression for the Product

Next, find the product of 2 and -4. This can be written as \[2 \times (-4)\].

Step 5: Simplify the Product

Now simplify the expression for the product: \[2 \times (-4) = -8\].

Step 6: Write the Division Expression

Now that you have both the sum and the product, write the entire expression for the division: \[\frac{-24}{-8}\].

Step 7: Simplify the Division

Finally, simplify the division: \[\frac{-24}{-8} = 3\].

Key Concepts

Arithmetic Operations

Arithmetic operations are the basic building blocks of math. They include addition, subtraction, multiplication, and division. In the given exercise, we need to use addition (or sum), multiplication (or product), and division.

First, let's review these operations in the context of the problem:

• Addition: Adding two numbers together. Example: \((-18) + (-6)\)
• Subtraction: Subtracting one number from another. Example: \((-18) - 6\)
• Multiplication: Multiplying two numbers. Example: \2 \times (-4)\
• Division: Dividing one number by another. Example: \frac{-24}{-8}\

Understanding these operations is crucial, as they are often combined to solve complex problems.

Expression Simplification

Expression simplification is the process of making an expression simpler and easier to understand. In the exercise, we started with the phrase: 'The sum of -18 and -6, divided by the product of 2 and -4'.

The simplification was done in steps:

• First, we simplified the sum \((-18) + (-6) = -24\)
• Next, we simplified the product \2 \times (-4) = -8\
• Finally, we simplified the division \frac{-24}{-8} = 3\

As you can see, breaking down the expression into steps makes the problem easier to solve.

Division

Division is an arithmetic operation where one number is divided by another. In this exercise, after finding the sum and product, we needed to perform division.

In mathematical notation, division is represented by the symbol \('/'\) or fraction bar \('\frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}}'\). For example, \frac{-24}{-8}\ can be read as '-24 divided by -8'.

Division is crucial when solving problems that involve multiple steps like this exercise. The key is to simplify the numbers as much as possible before performing division.

Here, \frac{-24}{-8}\ simplified to 3 because dividing two negative numbers results in a positive number:
\frac{{-24}}{{-8}} = \frac {24}{8} = 3\

Always remember, dividing negative by negative gives a positive result, dividing positive by negative (or vice-versa) gives a negative result.